Editing Linear trend estimation (section)

==Trends in clinical data==
Medical and [[biomedical]] studies often seek to determine a link between sets of data, such as of a clinical or scientific metric in three different diseases. But data may also be linked in time (such as change in the effect of a drug from baseline, to month 1, to month 2), or by an external factor that may or may not be determined by the researcher and/or their subject (such as no pain, mild pain, moderate pain, or severe pain). In these cases, one would expect the effect test statistic (e.g., influence of a [[statin]] on levels of [[cholesterol]], an [[analgesic]] on the degree of pain, or increasing doses of different strengths of a drug on a measurable index, i.e. a dose - response effect) to change in direct order as the effect develops. Suppose the mean level of cholesterol before and after the prescription of a statin falls from 5.6 [[mmol/L]] at baseline to 3.4 mmol/L at one month and to 3.7 mmol/L at two months. Given sufficient power, an [[Analysis of variance| ANOVA (analysis of variance)]] would most likely find a significant fall at one and two months, but the fall is not linear. Furthermore, a post-hoc test may be required. An alternative test may be a repeated measures (two way) ANOVA or [[Friedman test]], depending on the nature of the data. Nevertheless, because the groups are ordered, a standard ANOVA is inappropriate. Should the cholesterol fall from 5.4 to 4.1 to 3.7, there is a clear linear trend. The same principle may be applied to the effects of allele/[[genotype frequency]], where it could be argued that a [[single-nucleotide polymorphism]] in nucleotides XX, XY, YY are in fact a trend of no Y's, one Y, and then two Y's.<ref name=":1" />

The mathematics of linear trend estimation is a variant of the standard ANOVA, giving different information, and would be the most appropriate test if the researchers hypothesize a trend effect in their test statistic. One example is levels of serum [[trypsin]] in six groups of subjects ordered by age decade (10–19 years up to 60–69 years). Levels of trypsin (ng/mL) rise in a direct linear trend of 128, 152, 194, 207, 215, 218 (data from Altman). Unsurprisingly, a 'standard' ANOVA gives ''p''&nbsp;<&nbsp;0.0001, whereas linear trend estimation gives ''p''&nbsp;=&nbsp;0.00006. Incidentally, it could be reasonably argued that as age is a natural continuously variable index, it should not be categorized into decades, and an effect of age and serum trypsin is sought by correlation (assuming the raw data is available). A further example is of a substance measured at four time points in different groups:
{| class="wikitable"
|+
!#
!mean
!SD
|-
|1
|1.6
|0.56
|-
|2
|1.94
|0.75
|-
|3
|2.22
|0.66
|-
|4
|2.40
|0.79
|}
This is a clear trend. ANOVA gives ''p''&nbsp;=&nbsp;0.091, because the overall variance exceeds the means, whereas linear trend estimation gives ''p''&nbsp;=&nbsp;0.012. However, should the data have been collected at four time points in the same individuals, linear trend estimation would be inappropriate, and a two-way (repeated measures) ANOVA would have been applied.